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Improvement in growth of plants under the effect of magnetized water

  • Received: 28 August 2022 Revised: 30 September 2022 Accepted: 20 October 2022 Published: 25 November 2022
  • The magnetic field can change the polarity characteristics and hydrogen-bond structure of water; therefore, magnetized water can affect plant growth and development. Magnetized water is hexagonal water created by passing water through a specific magnet that can activate and ionize water molecules to change its structure. This review highlights the use of magnetized water in the agricultural sector to enhance plant growth and food productivity. We discussed the impact of magnetized water on seed germination, vegetative growth, fruit production, soil and pigments of treated plants. Plant growth and development can be improved both qualitatively and quantitatively via irrigation with magnetized water. It can promote seed germination, seedling early vegetative development, improvement of the mineral content of fruits and seeds, the enzyme activity of the soil, improved water use efficiency, higher nutrient content, and better transformation and consumption efficiency of nutrients; it can also mitigate soil salinity. Furthermore, magnetized water had a substantial good influence on the mobility and uptake of micronutrient concentrations, as well as promoted better growth criteria, all of which increased biomass and total yield. Also, irrigating plants with magnetized water resulted in a considerable increase in chloroplast pigments (carotenoids, chlorophyll a, and b) and photosynthetic activity. Magnetizing low-quality water (brackish water, saline water or water contaminated with metals) can be considered as an alternative tool to overcome the problem of scarcity and shortage of water resources. As a result, magnetic treatment of irrigation water could be a promising technique to boost agricultural production while also being environmentally beneficial in the future. The major challenge in using magnetized water in agriculture is creating pumps that are compatible with the technical and practical needs of magnetic systems while also effectively integrating irrigation components.

    Citation: Etimad Alattar, Eqbal Radwan, Khitam Elwasife. Improvement in growth of plants under the effect of magnetized water[J]. AIMS Biophysics, 2022, 9(4): 346-387. doi: 10.3934/biophy.2022029

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  • The magnetic field can change the polarity characteristics and hydrogen-bond structure of water; therefore, magnetized water can affect plant growth and development. Magnetized water is hexagonal water created by passing water through a specific magnet that can activate and ionize water molecules to change its structure. This review highlights the use of magnetized water in the agricultural sector to enhance plant growth and food productivity. We discussed the impact of magnetized water on seed germination, vegetative growth, fruit production, soil and pigments of treated plants. Plant growth and development can be improved both qualitatively and quantitatively via irrigation with magnetized water. It can promote seed germination, seedling early vegetative development, improvement of the mineral content of fruits and seeds, the enzyme activity of the soil, improved water use efficiency, higher nutrient content, and better transformation and consumption efficiency of nutrients; it can also mitigate soil salinity. Furthermore, magnetized water had a substantial good influence on the mobility and uptake of micronutrient concentrations, as well as promoted better growth criteria, all of which increased biomass and total yield. Also, irrigating plants with magnetized water resulted in a considerable increase in chloroplast pigments (carotenoids, chlorophyll a, and b) and photosynthetic activity. Magnetizing low-quality water (brackish water, saline water or water contaminated with metals) can be considered as an alternative tool to overcome the problem of scarcity and shortage of water resources. As a result, magnetic treatment of irrigation water could be a promising technique to boost agricultural production while also being environmentally beneficial in the future. The major challenge in using magnetized water in agriculture is creating pumps that are compatible with the technical and practical needs of magnetic systems while also effectively integrating irrigation components.



    The delay effect originates from the boundary controllers in engineering. The dynamics of a system with boundary delay could be described mathematically by a differential equation with delay term subject to boundary value condition such as [20]. There are many results available in literatures on the well-posedness and pullback dynamics of fluid flow models with delays especially the 2D Navier-Stokes equations, which can be seen in [1], [2], [8] and references therein. Inspired by these works, in this paper, we study the stability of pullback attractors for 3D Brinkman-Forchheimer (BF) equation with delay, which is also a continuation of our previous work in [6]. The existence and structure of attractors are significant to understand the large time behavior of solutions for non-autonomous evolutionary equations. Furthermore, the asymptotic stability of trajectories inside invariant sets determines many important properties of trajectories. The 3D Brinkman-Forchheimer equation with delay is given below:

    {utνΔu+αu+β|u|u+γ|u|2u+p=f(t,ut)+g(x,t),u=0, u(t,x)|Ω=0,u|t=τ=uτ(x), xΩ,uτ(θ,x)=u(τ+θ,x)=ϕ(θ), θ(h,0), h>0. (1)

    Here, (x,t)Ω×R+ with ΩR3 be a bounded domain with sufficiently smooth boundary Ω. u=(u1,u2,u3) is the velocity vector field, p is the pressure, ν>0 and α>0 denotes the Brinkman kinematic viscosity and the Darcy coefficients respectively, β>0 and γ>0 are the Forchheimer coefficients, the external force g(x,t)L2loc(R;H) is a locally square integrable function and the delay term is considered either

    (1). a general delay f(t,ut) with ut:[h,0]H defined as ut=u(t+s) which denotes constant, variable and distributed delays, see Caraballo and Real [1], [2].

    or

    (2). the special application of f(t,ut) as a sub-linear operator

    f(t,ut)=F(u(tρ(t))) (2)

    for a smooth function ρ() defined in Section 4, which satisfies subadditive and positive homogeneous property with second variable component, see Marín-Rubio and Real [8].

    The BF equation describes the conservation law of fluid flow in a porous medium that obeys the Darcy's law. The physical background of 3D BF model can be seen in [14], [9], [18], [19]. For the dynamic systems of problem (1) without delay, i.e., f(t,ut)=0, we can refer to [6], [10], [15], [16], [21] and literature therein for the existence of global weak solution and attractors. For the problem (1) with delay f(t,ut), the global existence of mild solution, continuous dependence on initial data and the minimal family of pullback attractors have been obtained in [6]. In addition, the upper semi-continuous property of pullback attractors as delay vanishes has been proved by virtue of some regular estimates. Furthermore, as a special application of the delay f(t,ut), the pullback dynamics of problem (1) with sub-linear operator (2) has also been shown. However, the asymptotic stability of trajectories inside pullback attractors is still open. Motivated by [3] and [17], applying regularity for weak solution and iteration technique with variable indices, we present some sufficient conditions with the generalized Grashof number to achieve the stability of pullback attractors in this paper. The main features and results can be summarized as follows.

    (a) For problem (1) with delay f(t,ut), we use the regular estimate to achieve an upper bound of Grashof number, which implies the exponential stability of trajectories inside pullback attractors. The proof does not depend on initial data with more regularity, see Section 3. Here we use, the delay f(t,ut) in problem (1) can be the constant, variable and distributed delays F(u(th)), F(u(tρ(t))) and 0hk(t,s)u(t+s)ds respectively, here F() is an appropriate function, see [1], [2].

    (b) For problem (1) with special application of f(t,ut) as the sub-linear operator, the variable indices have been introduced to deal with nonlinear term β|u|u+γ|u|2u and sub-linear operator by iterative argument, see Section 4.

    (c) The asymptotic stability of trajectories inside pullback attractors is further research of the results established in [6]. However, the stability of pullback attractors for (1) with infinite delay is still unknown.

    In this section, we give some notations and the equivalent abstract form of (1) in this section.

    Denoting E:={u|u(C0(Ω))3,divu=0}, H is the closure of E in (L2(Ω))3 topology, H and (,) denote the norm and inner product in H respectively. V is the closure of set E in (H1(Ω))3 topology, V and ((,)) denote the norm and inner product in V respectively. H and V are dual spaces of H and V respectively. Clearly, VHHV, H and V are dual spaces of H and V respectively, where the injection is dense, continuous. The norm denotes the norm in V, , be the dual product in V and V. Let P be the Helmholz-Leray orthogonal projection from (L2(Ω))3 onto the space H, we define A:=PΔ as the Stokes operator with domain D(A)=(H2(Ω))3V and λ is the first eigenvalue of A, the sequence {ωj}j=1 is an orthonormal system of eigenfunctions of A, and {λj}j=1 (0<λ1λ2) are eigenvalues of A corresponding to the eigenfunctions {ωj}j=1, see more details in [13].

    By the Helmholz-Leray projection defined above, (1) can be transformed to the abstract equivalent form

    {ut+νAu+P(αu+β|u|u+γ|u|2u)=Pf(t,ut)+Pg(t,x),u|Ω=0,u|t=τ=uτ(x),uτ(θ,x)=ϕ(θ,x) for θ(h,0), (3)

    then we show our results for (3) with f(t,ut) as either general case or its special case F(u(tρ(t))) in Sections 3 and 4, respectively.

    We also define some Banach spaces on delayed interval as CH=C([h,0];H), CV=C([h,0];V) with the norms

    ϕCH=supθ[h,0]ϕ(θ)H,  ϕCV=supθ[h,0]ϕ(θ)V,

    respectively. The Lebesgue integrable spaces on delayed interval can be denoted as LpH=Lp(h,0;H), LpV=Lp(h,0;V). The product space is defined as XH=C([τh,T];H)×C([τ,T];H) and MH=H×(CHL2V) for purpose of phase space in next sections.

    Some assumptions on the external forces and parameters which will be imposed in our main results are the following:

    (Hf) The function f:R×CHH satisfies:

    (a) For any ξCH, the function f(,ξ) is measurable and f(,0)0.

    (b) There exists a Lf>0 such that

    f(t,ξ)f(t,η)HLfξηCH, for ξ,ηCH.

    (c) There exists Cf>0, such that for all u,vC([τh,t];H)

    tτf(r,ur)f(r,vr)2HdrC2ftτhu(r)v(r)2Hdr, for τt. (4)

    (Hg) The function g(,)L2loc(R,V) satisfies that there exists η>0 such that

    teηsg(s,)2Vds<. (5)

    holds for any tτ.

    (H0) The coefficients satisfy α+3β+2γη2C2fαeηh0.

    Lemma 3.1. (The Gronwall inequality with differential form) Let m()C1[R+,R+], v(), h()C[R+,R+] and

    ddtm(t)v(t)m(t)+h(t), m(t=τ)=mτ, tτ. (6)

    Then

    m(t)mτetτv(s)ds+tτh(s)etsv(σ)dσds, tτ. (7)

    In this part, we shall present some retarded integral inequalities from Li, Liu and Ju [5]. Consider the following retarded integral inequalities:

    y(t)XE(t,τ)yτX+tτK1(t,s)ysXds+tK2(t,s)ysXds+ρ,  tτ, (8)

    where E, K1 and K2 are non-negative measurable functions on R2, ρ0 denotes a constant. Let X be a Banach space with spatial variable, based on the retarded Banach space above, then we use CX denotes the norm of space C([h,0];X) for some h0, y(t)0 is a continuous function defined on C([h,T];X), yt(s)=y(t+s) for s[h,0].

    Let L(E,K1,K2,ρ)={yC([h,T];X)|y0 and satisfies the inequality (8)}, and

    κ(K1,K2)=suptτ(tτK1(t,s)ds+tK2(t,s)ds).

    We assume that

    limt+E(t+s,s)=0 (9)

    uniformly with respect to sR+. Moreover, we suppose that κ(K1,K2)<+.

    Lemma 3.2. (The retarded Gronwall inequality) Denoting ϑ=suptsτE(t,s) and κ=κ(K1,K2), then we have the following estimates:

    (1) If κ<1, then for any R, ε>0, there exists ˜T>0 such that

    ytX<μρ+ε, (10)

    for t>˜T and all bounded functions yL(E,K1,K2,ρ) with y0XR, where μ=11κ.

    (2) If κ<11+ϑ, then there exist M, λ>0 which are independent on ρ such that

    ytXMy0Xeλt+γρ,  tτ (11)

    for all bounded functions yL(E,K1,K2,ρ), where γ=μ+11κc and c=max{ϑ1κ,1}.

    (3) If κ<11+ϑ, then the solution reduces to trivial for the occasion κc<1.

    Proof. See Li, Liu and Ju [5].

    Remark 1. (The special case: K2=0) Denote (K1,K2)=(K1,0) and let ϑ, κ, μ, γ be the constants defined in Lemma 3.2. Then we have the similar estimates as in Lemma 3.2.

    The minimal family of pullback attractors will be stated here in preparation for our main result.

    Some inequalities

    Lemma 3.3. (1) (See [7], [11]) Assume that β2, then for any a,bRn, we have

    (|a|β2a|b|β2b)(ab)γ0|ab|β,

    where γ0>0 is a constant which is determined by the volume of domain and its dimension, such as minγ0=126β2 in a 3-dimensional smooth domain.

    (2) The following Cq-inequality holds

    |xqyq|Cq(|x|q1+|y|q1)|xy|

    for the integer q2.

    Well-posedness

    Theorem 3.4. Assume that the external forces g(t,x) and f(t,ut) satisfy the hypothesis (Hg) and (Hf), the initial data (uτ,ϕ)MH=H×(CHL2V) and (H0) are also true. Then there exists a unique global weak solution u=u(,τ,uτ,ϕ)C([τh,T];H)L2(τ,T;V)L4(τ,T;L4(Ω)) of equation (1) on [τh,T].

    Proof. Step 1. Existence of local approximate solution.

    By the property of the Stokes operator A, the sequence of eigenfunctions {wi, i=1,2,} of the Stokes operator is an orthonormal complete basis of H formed by elements of V(H2(Ω))3 such that

    Awi=λiwi, i=1,2,. (12)

    Let Hm=span{w1,w2,,wm}, Pm:HHm be a projection, then the approximate solutions can be written as um(t)=mj=1hjm(t)wj (where hjm(t)=(vm(t),wj) is to be determined) which solve the problem

    {(tum,wj)+ν(um,wj)+(αum+β|um|um+γ|um|2um,wj)=(f(t,umt),wj)+g,wj,um(τ)=Pmuτ=uτm,umτ(θ,x)=Pmϕ(θ)=ϕm(θ) for θ[h,0], (13)

    Then it is easy to check that (13) is equivalent to an ordinary differential equations with unknown variable function hjm(t). By the Cauchy-Peano Theorem of ordinary differential equation, the problem (13) possesses a local solution over the time interval [0,tm].

    Step 2. Uniform estimates of approximate solutions.

    Multiplying (13) by hjm(t), and then summing from j=1 to m, it yields

    12ddtum2H+νum2V+αum2H+βum3L3(Ω)+γum4L4(Ω)|(g(t)+f(s,umt(s)),um)|αum2H+ν2um2V+12νg(t)2V+14αf(t,umt)2H. (14)

    Integrating in time, using the hypotheses on f(,) and g(t,x), by the Young inequality, we get

    um2H+νtτum2Vds+2βtτum3L3(Ω)ds+2γtτum4L4(Ω)dsuτ2H+C2f4α0hϕ(s)2Hds+12νtτg(s)2Vds+C2f4αtτum2Hds. (15)

    Using the Gronwall Lemma of integrable form, we conclude that

    {um}  is bounded in the spaceL(τ,T;H)L2(τh,T;V)L3(τ,T;L3(Ω))L4(τ,T;L4(Ω)).

    Step 3. Compact argument and passing to limit for deriving the global weak solutions.

    In this step, we shall prove {um} has a strong convergence subsequence by the Aubin-Lions Lemma along with the uniformly bounded estimate of dumdt in L2(0,T;V). By the estimates of um above step and continuous embedding VLp(Ω) with p[1,6] for three dimension, we can obtained that |um|umL2(τ,T;H) and |um|2umL2(τ,T;H). From the equation

    dumdt=νAumαumβ|um|umγ|um|2um+P(g(t)+f(t,umt) (16)

    and assumptions (Hf) and (Hg), we can see that {dum/dt} is bounded in L2(τ,T;V).

    By virtue of the Aubin-Lions Lemma, we obtain that {um} has a strong convergent subsequence (also denoted as {um} without confusion) with uL2(τh,T;V) and du/dtL2(τ,T;V) such that

    {um(t)u(t) weakly * in L(τ,T;H),um(t)u(t) stongly in L2(τ,T;H),um(t)u(t) weakly in L2(τ,T;V),dum/dtdu/dt weakly in L2(τ,T;V),f(,um)f(,u) weakly in L2(τ,T;H),umu(t) weakly in L3(τ,T;L3(Ω)),umu(t) weakly in L4(τ,T;L4(Ω)) (17)

    which coincides with the initial data um(τ)=Pmuτu(τ)=uτ and ϕm(s)ϕ(s).

    For the purpose of passing to limit in (13), denoting v=umu, we point out that we can deal with the nonlinear terms as the following novelty. Since wj is an eigenfunction of Stokes operator, we claim that

    Tτ(β|um|umβ|u|u,wj)dsCλ1βum4L4(τ,T;L4(Ω))umu4L4(τ,T;L4(Ω))+CβumuL(τ,T;H)u2L2(τh,T;H)

    and

    Tτ(γ|um|2umγ|u|2u,wj)dsCγum2L2(τ,T;V)umu4L4(τ,T;L4(Ω))+Cγumu4L4(τ,T;L4(Ω))(u2L2(τh,T;V)+um4L4(τ,T;L4(Ω))) (18)

    and the convergence of delayed external force f(t,umt) can be verified by the hypotheses.

    Thus, passing to the limit of (13), we conclude that u is at least one of global weak solutions for problem (1).

    The regularity

    Proposition 1. Assume that the external forces g(t) and f(t,ut) satisfy the hypothesis (Hg) and (Hf), the initial data (uτ,ϕ)MH=H×(CHL2V) and (H0) are also true. Then the global weak solution u in Theorem 3.4 has the regular boundedness in L(τ,T;V).

    Proof. Taking inner product of (3) with Au, it yields

    12ddtA1/2u2H+νAu2H+αA1/2u2H+βΩ|u|uAudx+γΩ|u|2uAudx=(f(t,ut),Au)+(g(t),Au). (19)

    According to Lemma 3.3, the nonlinear terms have the following estimates

    |β(|u|u,Au)|ν2Au2H+β4νu4L4 (20)

    and

    γΩ|u|2uAudx=γ2Ω|(|u|2)|2dx+γΩ|u|2|u|2dx (21)

    and

    (f(t,ut),Au)+(g(t),Au)12νf(t,ut)2H+12νg(t)2H+ν2Au2H, (22)

    hence, we conclude that

    ddtA1/2u2H+2αA1/2u2H+γΩ|(|u|2)|2dx+2γΩ|u|2|u|2dxβ2νu4L4+1νf(t,ut)2H+1νg(t)2H. (23)

    Letting t1st, neglecting the third and fourth terms on the left hand side of (23), integrating (23) with time variable from s to t, it yields

    A1/2u(t)2H+2αtsA1/2u(r)2HdrA1/2u(s)2H+β2νtsu(r)4L4dr+2νtsf(r,ur)2Hdr+2νtsg(r)2Hdr (24)

    and

    tsf(r,ur)2HdrL2fϕ(θ)2L2H+L2ftsu(r)2Hdr. (25)

    Then integrating with s from t1 to t, using the uniform boundedness of u in Theorem 3.4, we deduce that

    A1/2u(t)2Htt1A1/2u(s)2Hds+β2νtt1u(r)4L4dr+2L2fνϕ(θ)2L2H+2L2fνtτu(r)2Hdr+2νtt1g(r)2HdrC[ϕ2L2H+uτ2H]+Ctτg2Hds+2L2fνλ1tτu(r)2Vdr, (26)

    which means the uniform boundedness of the global weak solution u in L(τ,T;V). The proof has been finished.

    Uniqueness

    Proposition 2. Assume the hypotheses in Theorem 3.4 hold. Then the global weak solution u is unique.

    Proof. Using the same energy estimates as above, we can deduce the uniqueness easily, here we skip the details.

    To description of pullback attractors, the functional space MH=H×(CHL2V) is used as our phase space equipped with the norm (ξ,ζ)MH=ξH+ζL2V+ζCH for (ξ,ζ)MH. Based on the well-posedness, we shall verify the pullback dissipation and asymptotic compactness for the process to achieve the existence of pullback attractors, which also needs the following assumption:

    (H1) For every uL2(τh,T;V), there exists a η(0,νλ1) which is independent on u such that

    tτeηsf(s,us)2Hds<C2ftτheηsu(s)2Hds. (27)

    for any tT.

    The continuous process

    Proposition 3. For given f:R×CHH and gL2loc(R;V) satisfying (Hf),(Hg), (H1) and (H0). Then, the solution of problem (1) generates a biparametric family of mappings U(t,τ):MHMH by U(t,τ)(uτ,ϕ)=(u(t),ut), which is a continuous process.

    Pullback dissipation

    Lemma 3.5. Assume that f:R×CHH and gL2loc(R;V) satisfying (Hf),(Hg), (H1) and (H0). Then, for any (uτ,ϕ)MH, the solution u of (1) satisfies the estimates

    u(t)2He8ηCfα(tτ)(uτ2H+Cfϕ(r)2L2H)+e8ηCfαtνηλ1tτeηrg(r)2Vdr (28)

    and

    νtsu(r)2Vdru(s)2H+8Cfαus2L2H+1νtsg(r)2Vdr+8Cfαtsu(r)2Hdr, (29)
    βtsu(r)3L3(Ω)dru(s)2H+8Cfαus2L2H+1νtsg(r)2Vdr+8Cfαtsu(r)2Hdr, (30)
    γtsu(r)4L4(Ω)dru(s)2H+8Cfαus2L2H+1νtsg(r)2Vdr+8Cfαtsu(r)2Hdr. (31)

    Proof. By the energy estimate of (1) and using Young's inequality, we arrive at

    ddtu2H+2νu2V+2αu2H+2βu3L3(Ω)+2γu2L4(Ω)1νηλ1g2V+(νηλ1)u2V+2αu2H+8αf(t,ut)2H, (32)

    where η(0,νλ1).

    Multiplying the above inequality by eηt, we obtain

    ddt(eηtu2H)+eηtνλ1u2H+2βeηtu3L3(Ω)+2γeηtu2L4(Ω)1νηλ1eηtg2V+8Cfαeηtf(t,ut)2H.

    Thus integrating with respect to time variable, it yields

    eηtu2H+νλ1tτeηru(r)2Hdreητ(uτ2H+Cf0hϕ(r)2Hdr)+1νηλ1tτeηrg(r)2Vdr+8Cfαtτeηru(r)2Hdr (33)

    and by the Gronwall Lemma, we can derive the estimate in our theorem.

    Using the energy estimate of (1) again, we can check that

    ddtu2H+2νu2V+2αu2H+2βu3L3(Ω)+2γu2L4(Ω)1νg2V+νu2V+2αu2H+8αf(t,ut)2H, (34)

    Integrating from s to t, using the estimate of u in H, we can derive the desired result.

    Based on Lemma 3.5, we can present the pullback dissipation based on the following universes for the tempered dynamics.

    Definition 3.6. (Universe). (1) We will denote by DMHη the class of all families of nonempty subsets ˆD={D(t):tR}P(MH) such that

    limτ(eητsup(ξ,ζ)D(τ)(ξ,ζ)2MH)=0. (35)

    (2) DMHF denotes the class of families ˆD={D(t)=D:tR} with D a fixed nonempty bounded subset in MH.

    Remark 2. The universes DMHη and DMHη satisfy include closed property.

    Proposition 4. (The DMHη and DMHF pullback absorbing sets in MH) For given f:R×CHH and gL2loc(R;V) satisfying (Hf),(Hg), (H1) and (H0) holds. Then, the family ˆD0={D0(t):tR}MH is defined by

    D0(t)=¯BH(0,ρH(t))×(¯BL2V(0,ρL2H(t))¯BCH(0,ρCH(t)))

    is the pullback DMHη-absorbing set for the process U(t,τ) on MH and ˆD0DMHη, where the balls is defined as centered in the point zero and measured by the radius

    ρ2H(t)=1+e8ηCfα(th)νηλ1teηrg(r)2Vdr,ρ2L2V(t)=1ν[1+uτ2H+8Cfαϕ2L2H+g(r)2L2(th,t;V)ν+8Cfhαρ2H(t)].

    Moreover, the pullback DMHF-absorbing set can be defined as the same technique.

    Proof. Using the estimates in Lemma 3.5, choosing any ˆDDMHη(t), there exists a pullback time τ(ˆD,t)th such that

    u(t,τ;uτ,ϕ)2Hρ2H(t)=1+e8ηCfα(th)νηλ1teηrg(r)2Vdr (36)

    holds for any ττ(ˆD,t) and (uτ,ϕ)D(τ). Moreover, in particular, it yields that ut2CHρ2H(t). By the similar technique and estimate in Lemma 3.5, we derive that ut2L2Vρ2L2V(t). Combining the above estimate and the definition of universe, we conclude that ˆD0DMHη. The proof has been finished.

    Pullback asymptotic compactness

    Theorem 3.7. Assume that f:R×CHH and gL2loc(R;H) satisfying (Hf),(Hg), (H1) and (H0) holds. Then, the processes U(t,τ):MHMH generated by the solution of problem (1) is DMHη-pullback asymptotically compact.

    Proof. Step 1. Weak convergence of the sequence {un(t,x)} in the interval [th,t] for arbitrary tτ and weak convergence of {un(t)} in H.

    For arbitrary fixed tτ, consider a family ˆDDMHη, let {τn}(,t] with τn and {(uτn,ϕn)} with (uτn,ϕn)D(τn) be two sequences for all n, then we denote {(un,unt)}ˆD as a sequence with un()=u(;τn,uτn,ϕn).

    By using the similar energy estimate in Theorem 3.4 and technique in Proposition 4, there exists a pullback time τ(ˆD,t)t3h1, such that the sequence {un} with ττ(ˆD,t) is bounded in L(t3h1,t;H)L2(t2h1,t;V)L3(t2h1,t;L3(Ω))L4(t2h1,t;L4(Ω)). From the equation, we can check that

    (un)L2(th1,t;V)νunL2(th1,t;V)+αλ11unL2(th1,t;V)+βunL4(th1,t;L4(Ω))+Cλ1,|Ω|γunL2(th1,t;V)+Cαf(t,unt)L2(th1,t;H)+CνgL2(th1,t;V). (37)

    From the hypotheses (Hf), f(t,unt) is bounded in L2(th1,t;H), which implies {(un)} is bounded in L2(th1,t;V). Hence, by the Aubin-Lions Lemma and the diagonal procedure, there exists a subsequence (relabeled also as {un}) such that un(t)u(t) strongly in L2(th1,t;H). Combining the uniform boundedness of sequence above, it yields that

    {unu weakly * in L(t3h1,t;H),unu weakly in L2(t2h1,t;V),(un)u weakly in L2(th1,t;V),umu(t) weakly in L3(t2h1,t;L3(Ω)),umu(t) weakly in L4(t2h1,t;L4(Ω)),unu stongly in L2(th1,t;H),un(s)u(s) stongly in H, a.e. s(th1,t). (38)

    By Theorem 3.4, from the hypothesis on f, it follows that

    f(,un)f(,u) weakly in L2(th1,t;H). (39)

    Thus, from (38) and (39), we can conclude that uC([th1,t];H) is a weak solution for problem (1) with the initial data of u(,x) at the initial time th1 denoted as uth1.

    From the uniform bounded estimate of un by Proposition 1 in L(th1,t;V) and (un) is uniform bounded in L2(th1,t;V), using the Aubin-Lions-Simon Lemma (see [12]), we can derive that

    unu strongly in C([th1,t];H). (40)

    Therefore, we can conclude that

    un(sn)u(s)  weakly in H (41)

    for any {sn}[th1,t], sns[th1,t], which implies

    lim infnun(sn)Hu(s)H. (42)

    Step 2. The strong convergence of corresponding sequences via energy equation method: un(sn)u(s) strongly in C([th,t];H).

    The asymptotic compactness of sequence un in H will be presented in sequel, i.e.,

    un(sn)u(s)H0 as n+, (43)

    which is equivalent to prove (42) combining with

    lim supnun(sn)Hu(s)H (44)

    for a sequence {sn}[th,t] and sns as n+, which will be proved next.

    Using the energy estimate to all un and u, we obtain that for all th1s1s2t,

    un(s2)2H+νs2s1un(r)2Vdr+2βs2s1un(r)3L4(Ω)dr+2γs2s1un(r)4L4(Ω)2C2fαs2s2unr2Hdr+8νs2s1g(r)2Vdr (45)

    and

    u(s2)2H+νs2s1u(r)2Vdr+2βs2s1u(r)3L4(Ω)dr+2γs2s1u(r)4L4(Ω)2C2fαs2s2ur2Hdr+8νs2s1g(r)2Vdr. (46)

    Then, we define the functionals Jn(s) and J(s) defined for s[th1,t] as following

    Jn(s)=12un2Hsth1g(r),un(r)drsth1(f(r,unr),un(r))dr (47)

    and

    J(t)=12u(s)2Hsth1g(r),u(r)drsth1(f(r,ur),u(r))dr. (48)

    Combining the convergence in (38), observing that Jn(s) and J(s) are continuous and non-increasing in [th1,t], we derive that

    tth1g(r),un(r)dr2tth1g(r),u(r)dr (49)

    and

    tth1(f(r,unr),un(r))dr2tth1(f(r,ur),u(r))dr (50)

    as n+, which implies that

    Jn(s)J(s)  a.e.s(th1,t), (51)

    i.e., for  ε>0, there exists a nkN, for all nnk and sk[th1,t], such that

    |Jn(sk)J(sk)|ε2. (52)

    Since J(s) is continuous and Jn(s) is uniformly continuous with respect to time s, then for any ε>0, there exists ˜nkN such that for the sequence {sk}[th1,t] with sks for all n˜nk,

    |J(sk)J(s)|ε2, (53)

    Choosing ˉnk=max{nk,˜nk}, then for all n>ˉnk, it yields that

    |Jn(sn)J(s)||Jn(sn)J(sn)|+|J(sn)J(s)|<ε. (54)

    Therefore, for any {sn}[th1,t], we have

    lim supnJn(sn)J(s), (55)

    which implies

    lim supnun(sn)Hu(s)H. (56)

    we conclude the strong convergence un(sn)u(s) in C([th,t];H).

    Step 3. The strong convergence: un(sn)u(s) strongly in L2(th,t;V).

    Combining the energy estimates in (45) and (46), noting the energy functionals Jn() and J(), using the convergence in (38), we can deduce the norm convergence

    un(s)L2(th,t;V)u(s)L2(th,t;V). (57)

    Hence jointing with the weak convergence in (38), we can derive that un(sn)u(s) strongly in L2(th,t;V).

    Step 4. The DMHη-pullback asymptotic compactness.

    By using the results from Steps 2 to 4 and noting the definition of universe, we can conclude that the processes is DMHη-pullback asymptotic compact in MH, which means the proof has been finished.

    Remark 3. Using the similar technique, we can derive the processes U(t,τ):MHMH generated by the solution of problem (1) is DMHF-pullback asymptotic compact.

    Theorem 3.8. Assume that f:R×CHH and gL2loc(R;H) satisfying (Hf),(Hg), (H1) and (H0) holds. Then, the process U(t,τ):MHMH generated by the solution of problem (1) possess the minimal pullback attractors ADMHη(t) and ADMHF(t) in MH, which satisfy the following relation

    ADMHF(t)ADMHη(t). (58)

    Proof. From Proposition 3, we observe that the process U(t,τ) is continuous in MH. The DMHF and DMHη pullback absorbing sets are established by Proposition 4. By Theorems 3.7 and Remark 3 give the DMHη and DMHF pullback asymptotic compactness of the processes. Using the existence theory of pullback attractors in [3] or [4], we can conclude our desired results.

    Based on the universes defined in Definition 3.6, the relation between ADMHF(t) ADMHη(t) holds easily.

    Definition 3.9. The pullback attractors is asymptotically stable if the trajectories inside attractor reduces to a single orbit as τ.

    Theorem 3.10. Assume that 2νλ1L2fα>0, the external forces gL2loc(R;H) and f(t,ut) satisfy the hypothesis (Hf), (Hg) and (H1), the initial data (ϕ,uτ)MH and (H0) holds. Then the trajectories inside pullback attractors ADMHη(t) is asymptotically stable if

    G(t)K0,

    where G2(t)=g2H|tν2λ1 is a generalized Grashof number for the fluid flow, and

    K0={[ν2λ1(2νλ1+α)]/[4C|Ω|β(L2fα2L2fα12L2fα+1α)]}1/2,

    here C|Ω|>0 is a constant which depends on the volume of Ω.

    Proof. Let u(t) and v(t) be two weak solutions of problem (3) with delay f(t,ut) which subject to initial data

    u(τ+θ)|θ[h,0]=ϕ(θ),   u|t=τ=uτ (59)

    and

    v(τ+θ)|θ[h,0]=˜ϕ(θ),  v|t=τ=˜uτ (60)

    respectively. Denoting

    (u,ut)=U(t,τ)(uτ,φ)  and  (v,vt)=U(t,τ)(˜uτ,˜φ) (61)

    as two trajectories inside the pullback attractors, letting w=u(t)v(t) and wt=utvt, then it is easy to check that w satisfies the following problem

    {wt+νAw+P(αw+β(|u|u|v|v)+γ(|u|2u|v|2v))=P(f(t,ut)f(t,vt)),w|Ω=0,w(t=τ)=uτ˜uτ,w(τ+θ)=ϕ(θ)˜ϕ(θ), θ[h,0]. (62)

    Taking inner product of (62) with w in H, using Poincaré's inequality and Lemma 3.3, it follows

    γ(|u|2u|v|2v,uv)γγ0uv4L4 (63)

    and

    12ddtw2H+νw2V+αw2H+γγ0w4L4|β(|u|u|v|v,w)|+|(f(t,ut)f(t,vt),w)|β(Ω|u|2|w|dx+Ω|w||v|2dx)+α2w2H+L2f2αwt2H
    β(u2L4+v2L4)w2H+α2w2H+L2f2αwt2HC|Ω|β(u2V+v2V)w2H+α2w2H+L2f2αwt2H. (64)

    Using the Poincaré inequality and Lemma 3.1, noting that if

    2νλ1+α2C|Ω|β(u2V+v2V)>0, (65)

    then we can obtain

    w2Hetτ[2C|Ω|β(u2V+v2V)(2νλ1+α)]ds[uτ˜uτ2H++L2fαtτets[2νλ1+α2C|Ω|β(u2V+v2V)]dσwt2Hds]. (66)

    Denoting

    E(t,τ)=etτ[2νλ1+α2C|Ω|β(u2V+v2V)]ds (67)

    and

    K1(t,s)=L2fαets[2νλ1+α2C|Ω|β(u2V+v2V)]dσ (68)

    and

    Θ=suptsτE(t,s),    κ(K1,0)=suptτtτK1(t,s)ds, (69)

    by virtue of Lemma 3.2, choosing κ(K1,0)<11+Θ, then there exists M>0 and λ>0, such that we can obtain the estimate

    wt2HMuτ˜uτ2Heλ(tτ). (70)

    Substituting (70) into (64), using Lemma 3.1 again, we can conclude the following estimate

    w2Huτ˜uτ2Hetτ[2νλ1+α2C|Ω|β(u2V+v2V)]ds+L2fαMuτ˜uτ2Heλ(tτ)tτets[2νλ1+α2C|Ω|β(u2V+v2V)]dσds. (71)

    From (70) and (71), if we fixed uτ and ˜uτ and let τ, then we can conclude that the trajectories inside pullback attractors reduce to a point, which implies the pullback attractors is asymptotically stable provided that

    2νλ1+α>2C|Ω|βu2V+v2Vt, (72)

    where

    ht=lim supτ1tτtτh(r)dr. (73)

    Since u and v are the global weak solutions for problem (3), we will use some delicate estimates to make (72) more explicit next. Multiplying (3) with u and integrating by parts over Ω, we have

    12ddtu2H+νA1/2u2H+αu2H+βu3L3+γu4L4αu2H+12α[f(t,ut)2H+g2H]αu2H+L2f2αut2H+12αg2H. (74)

    Using the Poincaré inequality and Lemma 3.1, then we can obtain

    u2He2νλ1(tτ)uτ2H++L2fαtτe2νλ1(ts)us2Hds+1αtτe2νλ1(ts)g2Hds. (75)

    Denoting

    E(t,τ)=e2νλ1(tτ) (76)

    and

    K1(t,s)=L2fαe2νλ1(ts) (77)

    and

    ρ=1αtτe2νλ1(ts)g2Hds, (78)

    letting

    Θ=suptsτE(t,s),    κ(K1,0)=suptτtτK1(t,s)ds, (79)

    by virtue of Lemma 3.2, choosing κ(K1,0)<11+Θ, then there exists ˆM>0 and ˆλ>0, such that we can obtain the estimate

    ut2HˆMuτ2Heλ(tτ)+2L2fα12L2fαtτe2νλ1(ts)g2HdsˆMuτ2Heλ(tτ)+2L2fα12L2fαtτg2Hds. (80)

    Substituting (80) into (75), using Lemma 3.1 again, we can conclude the following estimate

    u2HCuτ2Heλ(tτ)+(L2fα2L2fα12L2fα+1α)tτg2Hds. (81)

    Integrating (74) from τ to t, it follows

    u2H+2νtτu2Vds+2βtτu3L3ds+2γtτu4L4ds[1αϕ2L2H+uτ2H]+L2fαtτut(s)2Hds+1αtτg2Hds. (82)

    By the estimate of (80) and (81), we derive

    tτu(r)4L4drC[1αϕ2L2H+uτ2H]+(L2fα2L2fα12L2fα+1α)tτg2Hds (83)

    and

    tτu(r)2VdrC[1αϕ2L2H+uτ2H]+(L2fα2L2fα12L2fα+1α)tτg2Hds. (84)

    Combining (72), (73) with (84), we conclude that

    u2V+v2V|t2(L2fα2L2fα12L2fα+1α)g2H|t (85)

    and hence the asymptotic stability holds provided that

    4C|Ω|β(L2fα2L2fα12L2fα+1α)g2H|t2νλ1+α. (86)

    If we define the generalized Grashof number as G(t)=(g2H|tν2λ1)1/2, then a sufficient condition for the asymptotic stability of trajectories inside pullback attractors can be conclude as

    G(t){(2νλ1+α)/[4C|Ω|ν2βλ1(L2fα2L2fα12L2fα+1α)]}1/2=K0, (87)

    which completes the proof for our first result.

    Remark 4. Theorem 3.10 is a further research for the existence of pullback attractor in [6].

    We first state some hypothesis on the external forces and sub-linear operator.

    (Hf) The function f(t,ut)=F(u(tρ(t))) satisfies the following assumptions.

    (Hf)(a) The function ρC1([0,+);[0,h]), and there exists a constant ρ satisfying

    |dρdt|ρ<1,  t0.

    (Hf)(b) The external force f(,y):[τ,+)×HH is measurable for all yH and f(,0)=0. In addition, there exist functions a,b:[τ,+)[0,+), with aLqloc(R) and bL1loc(R) for all Tτ and 1q+ with lim supτtτb(s)ds=˜b0(0,+), such that

    F(y)2Ha(t)y2H+b(t),  tτ,yH. (88)

    (Hf)(c) There exist κL(τ,T), L(R)>0 and R>0, such that

    F(u)F(v)HL(R)κ12(t)uvH, u,vH. (89)

    holds for uHR,vHR and t[τ,T].

    (Hg) The function g(,)L2loc(R,H), which satisfies that there exists a m>0, such that

    temsg(s,)2Hds<,  tR. (90)

    (˜H0) When aLqloc(R), it holds

    ν2aLqloc(R)1ρ>0. (91)

    In this part, the well-posedness and pullback attractors for problem (1) with sub-linear operator will be stated for our discussion in sequel.

    Well-posedness

    Assume that the initial date uτH and ϕCHL2q(h,0;H) with 1q+1q=1 and recall that (1) with sub-linear operator has the following abstract form:

    {u(t)+tτP(νAu+αu+β|u|u+γ|u|2u)ds=u(τ)+tτP(F(u(sρ(s)))+g(s,x))ds,w|Ω=0,u(t=τ)=uτ,u(τ+t)=ϕ(t), t[h,0], (92)

    which possesses a global mild solution as the following theorem.

    Theorem 4.1. Assume that the external forces g(t) and f(t,ut)=F(u(tρ(t))) satisfy the hypothesis (Hf) and (Hg), the initial data uτH and ϕCHL2qH and (˜H0) are also true. Then there exists a unique global mild solution u=u(,τ,uτ,ϕ)L(τh,T;H)L2(τ,T;V)L4(τ,T;L4(Ω)) of problem (1) for special case of f(t,ut)=F(tρ(t)), such that it satisfies (92) in distributed sense and the following energy equality

    u(t)2H+2νtτu(s)2Vds+2αtτu(s)2Hds+2βtτu(s)3L3ds+2γtτu(s)4L4ds=uτ2H+2tτ[(F(u(sρ(s))),u(s))+2(g(s,x),u(s))]ds. (93)

    Moreover, we can define a continuous process {U(t,τ)|tτ,τR} as </italic><italic>U(t,τ):(CHL2qH)×HCH×H, i.e., U(t,τ)(ϕ,uτ)=(ut,u).

    Proof. Using the Galerkin method and compact argument as in Section 3.3, we can easily derive the result.

    The pullback dynamics

    After obtaining the existence of the global well-posedness, we establish the existence of the pullback attractors to (1) with sub-linear operator.

    Theorem 4.2. (The pullback attractors in H) Assume that (Hf) and (Hg) hold, the initial data uτH and ϕCHL2qH and (˜H0) are also true. then the process U(t,τ) associated to problem (1) with f(t,ut)=F(u(tρ(t))) has two families of pullback attractors ACH×H(t) similar as in Theorem 3.7.

    Proof. Using the similar technique as in Section3.3, we can obtain the existence of pullback attractors, here we skip the details.

    Theorem 4.3. We assume that the external forces g(t) and f(t,ut)=F(u(tρ(t))) satisfy the hypothesis (Hf) and (Hg), the initial data (ϕ,uτ)(CHL2qH)×H and (˜H0) holds true.

    Then the trajectories inside pullback attractors ACH is asymptotically stable if

    G(t)˜K0, (94)

    where G2(t)=g2H|tν2λ1 is defined as the generalized Grashof number for the fluid flow, g2H|t=limτ1tτtτg(s)2Hds and

    ˜K0={(2νλ1+α)/[2C|Ω|βνλ1(1α2(1ρ)+˜a(t)L1)]}1/2>0,

    here C|Ω|>0 is a constant dependent on the volume of Ω.

    Proof. Step 1. The inequality for asymptotic stability of trajectories.

    Let u(t) and v(t) be two solutions of problem (92) with initial data

    u(θ+τ)|θ[h,0]=ϕ(θ)|θ[h,0],  u|t=τ=uτ (95)

    and

    v(θ+τ)|θ[h,0]=˜ϕ(θ)|θ[h,0],  v|t=τ=˜uτ (96)

    respectively, then u,v can be represented by

    (u,ut)=(U(t,τ)uτ,U(t,τ)ϕ),  (v,vt)=(U(t,τ)˜uτ,U(t,τ)˜ϕ). (97)

    If we denote w=u(t)v(t) and w(tρ(t))=u(tρ(t))v(tρ(t)), then it is easy to check that w satisfies the following problem

    {wt+νAw+P(αw+β(|u|u|v|v)+γ(|u|2u|v|2v))=P(F(u(tρ(t)))F(v(tρ(t)))),w|Ω=0,w(t=τ)=uτ˜uτ,w(τ+θ)=ϕ(θ)˜ϕ(θ), θ[h,0]. (98)

    Multiplying (98) with w and integrating by parts in Ω, using the Poincaré and Young inequalities, from (63) we obtain that

    12ddtw2H+νw2V+αw2H+γγ0w4L4|β(|u|u|v|v,w)|+|(F(u(tρ(t)))F(v(tρ(t))),w)|C|Ω|β(u2V+u2V)w2H+α2w2H+1αF(u(tρ(t)))F(v(tρ(t)))2HC|Ω|β(u2V+u2V)w2H+α2w2H+L2(R)κ(t)αw(tρ(t))2H. (99)

    Using the Poincaré inequality and Lemma 3.1, noting that if

    2νλ1+α2C|Ω|β(u2V+v2V)>0, (100)

    then we can obtain

    w2Hetτ[2C|Ω|β(u2V+v2V)(2νλ1+α)]ds[uτ˜uτ2H++L2(R)κ(t)Lαtτets[2νλ1+α2C|Ω|β(u2V+v2V)]dσ×w(tρ(t))2Hds]. (101)

    Denoting

    E(t,τ)=etτ[2νλ1+α2C|Ω|β(u2V+v2V)]ds (102)

    and

    K1(t,s)=L2(R)κ(t)Lαets[2νλ1+α2C|Ω|β(u2V+v2V)]dσ (103)

    and

    Θ=suptsτE(t,s),    κ(K1,0)=suptτtτK1(t,s)ds, (104)

    by virtue of Lemma 3.2, choosing κ(K1,0)<11+Θ, then there exists ˜M>0 and ˜λ>0, such that we can obtain the estimate

    w(tρ(t))2H˜Muτ˜uτ2He˜λ(tτ). (105)

    Substituting (105) into (99), using Lemma 3.1 again, we can conclude the following estimate

    w2Huτ˜uτ2Hetτ[2νλ1+α2C|Ω|β(u2V+v2V)]ds+L2(R)κ(t)Lα˜Muτ˜uτ2He˜λ(tτ)×tτets[2νλ1+α2C|Ω|β(u2V+v2V)]dσds. (106)

    From the result in last section, we can find that the pullback attractors is asymptotically stable as τ provided that

    2νλ1+α>2C|Ω|βu2V+v2Vt. (107)

    Step 2.Some energy estimate for (1) with sub-linear operator.

    Multiplying (3) with u in H, and then integrating by parts over Ω, we have

    12ddtu2H+νA1/2u2H+αu2H+βu3L3+γu4L4αu2H+12α[f(t,u(tρ(t)))2H+g2H]. (108)

    Moreover, let θ=sρ(s), then it yields

    dθ=(1ρ(s))ds, a(t)˜a(ˉt)Lp(τ,T), (109)

    which means ˜aLq(τh,τ) and

    tτf(s,u(sρ(s)))2Hdstτa(s)u(sρ(s))2Hds+Tτb(s)ds 11ρtρ(t)τρ(τ)˜a(s)u(s)2Hds+tτb(s)ds 11ρ(0ρ(τ)˜a(t+τ)ϕ(t)2Hdt+tτ˜a(s)u(s)2Hds)+tτb(s)ds11ρ(ϕ(t)2L2qH˜aLq(τh,τ)+tτ˜a(s)u(s)2Hds)+tτb(s)ds, (110)

    Integrating (108) with time variable from τ to t, we conclude that

    u2H+2νtτu2Vds+2βtτu3L3ds+2γtτu4L4ds˜aLq(τh,τ)α(1ρ)ϕ(t)2L2qH+uτ2H+1α(1ρ)tτ˜a(s)u(s)2Hds+1αtτg2Hds+1αtτb(s)ds, (111)

    then we can achieve that

    u(t)2H[˜aLq(τh,τ)α(1ρ)ϕ(t)2L2qH+uτ2H]eχσ(t,τ)+1αtτg2Heχσ(t,s)ds+1αtτb(s)eχσ(t,s)ds, (112)

    where the new variable index χσ(,) is introduced as

    χσ(t,s)=(2νλ1σ)(ts)1α(1ρ)ts˜a(r)dr, (113)

    which satisfies the relations

    χσ(0,t)χσ(0,s)=χσ(t,s) (114)

    and

    χσ(0,r)χσ(0,t)+(2νλ1δ)h,  if 2νλ1+αδ>0 (115)

    for r[th,t].

    Moreover, using the variable index introduced above, we can conclude that

    2νtτu(r)2Vdr˜aLq(τh,τ)α(1ρ)ϕ(t)2L2qH+uτ2H+1αtτg2Hds+1αtτb(s)ds+1α(1ρ)[˜aLq(τh,τ)α(1ρ)ϕ(t)2L2qH+uτ2H]tτ˜a(s)eχσ(s,τ)ds+1α2(1ρ)tτg(s)2Hdstτ˜a(s)ds+bL1(τ,T)α2(1ρ)tτ˜a(s)ds. (116)

    Step 3. The sufficient condition for asymptotic stability of trajectories inside pullback attractors.

    Combining (107) with (116), we conclude that

    2C|Ω|βu2V+v2V|t2C|Ω|βν[(1α2(1ρ)+tτ˜a(s)ds)g(t)2H|t+1αb(t)L1|t+bL1(τ,T)α2(1ρ)˜a(t)L1|t]. (117)

    and hence the asymptotic stability holds provided that

    (1α2(1ρ)+˜a(t)L1)g(t)2H|t+1αb(t)L1|t+bL1(τ,T)α2(1ρ)˜a(t)L1|tν(2νλ1+α)2C|Ω|β. (118)

    If we define the generalized Grashof number as G(t)=(g2H|tν2λ1)1/2, neglecting the positive terms on the left-hand side of (118), then a sufficient condition for the asymptotic stability of trajectories inside pullback attractors can be conclude as

    G(t){(2νλ1+α)/[2C|Ω|βνλ1(1α2(1ρ)+˜a(t)L1)]}1/2=˜K0, (119)

    which completes the proof for our first result.

    Remark 5. If we denote

    lim supτ1tτtτb(r)dr=b0[0,+) (120)

    and

    lim supτ1tτtτ˜a(r)dr=˜a0[0,+), (121)

    such that there exists some σ>0, the following assumption

    ν(2νλ1+α)2C|Ω|β>b0α+bL1(τ,T)˜α0α2(1ρ)+δ (122)

    holds. Then more precise sufficient condition for the asymptotic stability of pullback attractors is

    G(t)[ν(2νλ1+α)2C|Ω|βb0αbL1(τ,T)˜α0α2(1ρ)ν2λ1(1α2(1ρ)+˜a(t)L1)]1/2 (123)

    which has smaller upper boundedness than (119).

    The structure and stability of 3D BF equations with delay are investigated in this paper. A future research in the pullback dynamics of (1) is to study the geometric property of pullback attractors, such as the fractal dimension.

    Xin-Guang Yang was partially supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003). Xinjie Yan was partly supported by Excellent Innovation Team Project of "Analysis Theory of Partial Differential Equations" in China University of Mining and Technology (No. 2020QN003). Ling Ding was partly supported by NSFC of China (Grant No. 1196302).

    The authors want to express their most sincere thanks to refrees for the improvement of this manuscript. The authors also want to thank Professors Tomás Caraballo (Universidad de Sevilla), Desheng Li (Tianjin University) and Shubin Wang (Zhengzhou University) for fruitful discussion on this subject.



    Conflict of interest



    All authors declare no conflict of interest regarding this paper.

    Author contribution



    All authors contributed to the conception and design of the manuscript. All authors critically revised the manuscript and approved the final manuscript. Etimad Alattar led the conceptualization, analysis, interpretation and writing of the manuscript. Eqbal Radwan contributed to writing the first draft of the manuscript, prepared the tables and analyzed the data. Khitam Elwasife contributed to the revision and editing of the manuscript.

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